NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS
SINGULARITIES
J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
Abstract. We consider a weighted homogeneous germ of complex analytic variety
(X, 0) ⊂ (Cn , 0) and a function germ f : (Cn , 0) → (C, 0). We derive necessary and
sufficient conditions for some deformations to have non-negative degree (i.e., for any
additional term in the deformation, the weighted degree is not smaller) in terms of an
adapted version of the relative Milnor number. We study the cases where (X, 0) is an
isolated hypersurface singularity and the invariant is the Bruce-Roberts number of f
with respect to (X, 0), and where (X, 0) is an isolated complete intersection or a curve
singularity and the invariant is the Milnor number of the germ f : (X, 0) → C. In the
last part we give some formulas for the invariants in terms of the weights and the degrees
of the polynomials.
1. Introduction
The Milnor number is an important invariant in Singularity Theory. When it is defined,
it reflects interesting geometric properties of the germ. For instance, if f : (Cn , 0) → C
is an isolated singularity function germ, then its generic fiber has the homotopy type of
a bouquet of spheres of real dimension n − 1. The number of such spheres, denoted by
µ(f ), is the Milnor number of f (see [15]).
Lê-Ramanujam ([12]) and Timourian ([23]) show that a family of hypersurfaces in
(Cn , 0) (with n 6= 3) has constant Milnor number if and only if all the hypersurfaces
in the family have the same topological type. In other words, the Milnor number is
a topological invariant whose constancy in a family controls the topological triviality.
Because of these facts it is important to know whether a family has constant Milnor
number. In [24], Varchenko shows the following result.
Theorem 1.1. If f : (Cn , 0) → C is a weighted homogeneous polynomial with an isolated
P
singularity and if ft (x) = f (x) + ki=1 σi (t)αi (x) is a deformation of f with σi (0) = 0,
then the Milnor number of ft , µ(ft ), is constant if and only if the weighted degree of αi is
not smaller than the weighted degree of f , for all i = 1, . . . , k.
A deformation like the one in theorem 1.1 (the weighted degree of αi is not smaller
than the weighted degree of f , for all i such that σi is not zero) is called a non-negative
deformation.
In this paper, we want to find necessary and sufficient conditions for a deformation to
be non-negative in other cases where the Milnor number (or another similar invariant)
is well defined. In the case of varieties with isolated singularity, the Milnor number is
2000 Mathematics Subject Classification. Primary 32S15; Secondary 58K60, 32S30.
Key words and phrases. Milnor number, Bruce-Roberts number, non-negative deformations.
The first author was partially supported by DGICYT Grant MTM2012–33073 and CAPES-PVE Grant
88881.062217/2014-01. The second author was partially supported by FAPESP Grants 2011/08877-3 and
2013/14014-3. The third author was partially supported by FAPESP Grant 2013/10856-0 and CNPq
Grant 309626/2014-5.
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J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
known to be well defined for a space curve singularity [5], an isolated complete intersection
singularity - ICIS [11], or an isolated determinantal singularity - IDS [19]. In the case of
functions, on one hand we have the Bruce-Roberts number of a function f : (Cn , 0) → C
with respect to a variety (X, 0) ⊂ (Cn , 0) [3]. On the other hand, we can also consider
the Milnor number of a function f : (X, 0) → C, where (X, 0) is either a curve [9, 17] or
an IDS [19]. Moreover, we also provide some formulas for these invariants in terms of the
weights and the degrees of the polynomials.
2. Weighted homogeneous functions and varieties
We use this section to present the basic definitions we will use in the work. Let us
denote by On the local ring of germs of complex analytic functions f : (Cn , 0) → C. We
fix positive integer numbers w1 , . . . , wn ∈ N∗ such that gcd(w1 , . . . , wn ) = 1. Assume that
f ∈ On is expressed as a convergent power series:
X
f (x) =
aα x α
α∈Nn
We say that f is weighted homogeneous if there exists d such that α1 w1 + · · · + αn wn = d
for all α such that aα 6= 0. In such a case, we say that d is the weighted degree of f and
we write, wt(f ) = d. Equivalently, f is a polynomial such that
f (tw1 x1 , . . . , twn xn ) = td f (x1 , . . . , xn ),
∀x ∈ Cn , ∀t ∈ C.
We say that a variety germ (X, 0) ⊂ (Cn , 0) is weighted homogeneous if it is the zero
set of an ideal I ⊂ On which can be generated by weighted homogeneous germs φ1 , . . . , φp
(with respect to the same weights, but with possibly different degrees). Now, this is
equivalent to say that
(tw1 x1 , . . . , twn xn ) ∈ X,
∀x ∈ X, ∀t ∈ C.
In particular, we take the convention that (Cn , 0) is itself weighted homogeneous. Associated with (X, 0) we can consider in a natural way the ring
C[x1 , . . . , xn ]/ hφ1 , . . . , φp i
as a graded ring with respect to the weights (w1 , . . . , wn ) (see, for instance, [14, p. 93]).
Definition 2.1. Let (X, 0) ⊂ (Cn , 0) be weighted homogeneous as above. We say that
a polynomial germ f : (X, 0) → C is weighted homogeneous if it is homogeneous in the
graded ring C[x1 , . . . , xn ]/ hφ1 , . . . , φp i. In particular, there exists d ∈ N such that
f (tw1 x1 , . . . , twn xn ) = td f (x1 , . . . , xn ),
∀x ∈ X, ∀t ∈ C.
In this case, we call d the weighted degree of f and write d = wt(f ).
Any non-zero polynomial germ f : (X, 0) → C can be written in a unique way as:
f = fd + fd+1 + · · · + fl ,
with fd 6= 0, where each fi is weighted homogeneous of degree i. We call fd the initial
part of f and denote it by in(f ).
We say that f is semi-weighted homogeneous if in(f ) has isolated singularity (that is,
there exist a representative X of (X, 0) such that X \ {0} is non-singular and in(f ) is
regular on X \ {0}).
NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES
3
In this paper, by a deformation of a map germ f : (X, 0) → (Y, 0) we mean another
map germ F : (C × X) → (Y, 0) such that F (0, x) = f (x), for all x ∈ X. We also assume
that F is origin preserving, that is, F (t, 0) = 0 for all t, so we have a 1-parameter family of
map germs ft : (X, 0) → (Y, 0) given by ft (x) = F (t, x). By abuse of notation, sometimes
it is usual to denote the deformation F just by ft . On the other hand, we can also consider
the associated unfolding F̃ : (C × X) → (C × Y, 0) given by F̃ (t, x) = (t, ft (x)).
In the particular case of a polynomial function f : (X, 0) → (C, 0), any polynomial
deformation ft can be written as:
(1)
ft (x) = f (x) +
k
X
σi (t)αi (x),
i=1
for some αi : (X, 0) → (C, 0) and σi : (C, 0) → (C, 0) i = 1, . . . , k.
Definition 2.2. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous variety germ. We
say that a polynomial deformation ft of f : (X, 0) → (C, 0) as in (1) is non-negative if
wt(in(αi )) ≥ wt(in(f )), for i = 1, . . . , k such that αi 6= 0.
3. The Bruce-Roberts number
Let (X, 0) ⊂ (Cn , 0) be an analytic variety germ. Bruce and Roberts introduce in [3]
a generalization of µ(f ) which we call Bruce-Roberts number of f with respect to (X, 0).
It is defined by
On
µBR (f, X) = dimC
,
df (θX )
where θX is the On -module of vector fields in (Cn , 0) which are tangent to (X, 0) and df
denotes the differential map of f . We remark that if X = ∅, then µBR (f, X) = µ(f ).
Like the Milnor number of f , this number shows some geometric properties of f and X.
For instance, if we consider the group RX of biholomorphisms of (Cn , 0) which preserve
X, then f is finitely determined with respect to the action of RX on On if and only if
µBR (f, X) is finite and, in this case, the codimension of the orbit of f under this action
is equal to µBR (f, X) (see [3]).
In [18], we show the following result, relating the Milnor and Bruce-Roberts numbers.
Theorem 3.1. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous hypersurface with an
isolated singularity and let f : (Cn , 0) → C be an RX -finitely determined function germ.
Then,
µBR (f, X) = µ(f ) + µ(X ∩ f −1 (0), 0).
The goal of this section is to provide a result like theorem 1.1 for the Bruce-Roberts
number. In order to do this we will need the following related property. We recall that a
deformation ft of a function germ f : (Cn , 0) → (C, 0) is called C 0 -RX -trivial if there is a
homeomorphism Ψ : (C × Cn ) → (C × Cn ) which is an unfolding of the identity map in
Cn such that ft ◦ ψt = f and ψt preserves (X, 0) (i.e., ψt (X) = X for all t ∈ C).
Theorem 3.2. [6, 21] Let (X, 0) be a weighted homogeneous variety with an isolated
singularity and let f be a weighted homogeneous RX -finitely determined function. If ft is
a non-negative deformation of f , then ft is C 0 -RX -trivial.
Remark 3.3. In [6, 21] it is used the term “consistent with (X, 0)” to refer to the fact
that f is weighted homogenous with the same weights of (X, 0), but here this is not
necessary since we fix the weights from the beginning.
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J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
An interesting open problem is to know whether the Bruce-Roberts number is or is
not a topological invariant. A partial answer is given in [8]: if (X, 0) is a hypersurface
whose logarithmic characteristic variety, LC(X), is Cohen-Macaulay and if ft is a C 0 RX -trivial deformation of f , then µBR (ft , X) is constant (see [22] for the definition of
LC(X)). In [18], we see that if (X, 0) is a weighted homogeneous hypersurface with an
isolated singularity, then LC(X) is Cohen-Macaulay. Therefore, we have:
Corollary 3.4. Let (X, 0) be a weighted homogeneous hypersurface with an isolated singularity and let f : (Cn , 0) → C be an RX -finitely determined function germ. If ft is a
C 0 -RX -trivial deformation of f , then µBR (ft , X) is constant.
In [1], it is shown that if (X, 0) is a weighted homogeneous hypersurface with an isolated singularity and if f and g are C 0 -RX -equivalent function germs then µBR (f, X) =
µBR (g, X). The following example shows that the converse of this result is not true.
Example 3.5. Let (X, 0) = (φ−1 (0), 0) ⊂ (C2 , 0), where φ(x, y) = x2 + y 3 . Let f, g :
(C2 , 0) → C defined by f (x, y) = x3 + y 3 and g(x, y) = x2 + y 5 .
We have
µBR (f, X) = µBR (g, X) = 9,
0
but f and g are not C -RX -equivalent (in fact, they are not C 0 -R-equivalent, since the
plane curve f −1 (0) has three branches while g −1 (0) has only one branch).
However, inspired by theorem 1.1, it is natural to ask if the Bruce-Roberts number is
invariant under non-negative deformations. We present a partial answer in the following
result.
Theorem 3.6. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous hypersurface with isolated
singularity, let f : (Cn , 0) → C be a weighted homogeneous RX -finitely determined function and let ft be a deformation of f as in (1). The following statements are equivalent:
(1) the family ft is C 0 -RX -trivial;
(2) ft is a non-negative deformation;
(3) µBR (ft , X) is constant.
Proof. The implication (2) ⇒ (1) is given in theorem 3.2, so we see (1) ⇒ (2). If ft is
C 0 -RX -trivial, then it is C 0 -R-trivial and thus, µ(ft ) is constant (since ft has an isolated
singularity). By theorem 1.1, ft is a non-negative deformation.
The implication (1) ⇒ (3) is given in corollary 3.4, so it is enough to show (3) ⇒ (2).
If µBR (ft , X) is constant for t small enough, then, from theorem 3.1 and from the fact
that µ(ft ) and µ(X ∩ ft−1 (0), 0) are upper semicontinuous, these numbers are constant.
Therefore, the result follows from theorem 1.1.
We see that the implication (3) ⇒ (2) is also true even if f is weighted homogeneous
with different weights of (X, 0) (that is, it is not “consistent with (X, 0)” in the terminology
of [6, 21]). However, the converse is not true in general in that case, as the following
example shows.
Example 3.7. Let (X, 0) be the homogeneous surface germ defined by the zeros of φ =
x2 + y 2 + z 2 . Let f : (C3 , 0) → C be the function germ defined by
f (x, y, z) = z 5 + y 7 x + x15 ,
then f is weighted homogeneous. Finally, we consider the non-negative deformation ft of
f given by
ft (x, y, z) = f (x, y, z) + ty 6 z.
NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES
5
We have µBR (f, X) = 429, but µBR (ft , X) = 425 for t 6= 0 small enough.
4. The Milnor number of a function on a curve and on an ICIS
Let (X, 0) ⊂ (Cn , 0) be a reduced curve germ. We denote its local ring by OX,0 . We also
denote by ΩX,0 the module of holomorphic 1-forms on (X, 0) and by ωX,0 the dualizing
module of Grothendieck (see [4, Chapter 3]); that is,
n−1
(OX,0 , ΩnCn ,0 ),
ωX,0 := ExtO
n
where ΩnCn ,0 is the module of holomorphic n-forms on (Cn , 0). We consider a normalization
of the curve n : (X, 0) → (X, 0), which induces the so-called class map cX,0 : ΩX,0 → ωX,0
as the composition
ΩX,0 → ΩX,0 ∼
= ωX,0 → ωX,0 .
The composition of the differential operator d : OX,0 → ΩX,0 with the class map gives a
map which we also denote by d : OX,0 → ωX,0 . The Milnor number of the reduced curve
(X, 0) is defined in [5] as
ωX,0
µ(X, 0) = dimC
.
dOX,0
Let f : (X, 0) → C be a finite function germ on the curve X. The Milnor number µ(f )
has been introduced in [9] for curves in C3 and later in [17] for the general case as
(2)
µ(f ) = dimC
ωX,0
,
df ∧ OX,0
where df ∧ : OX,0 → ωX,0 denotes the composition of df ∧ : OX,0 → ΩX,0 with the class
map cX,0 : ΩX,0 → ωX,0 . In [20], it is shown that
(3)
µ(f ) = µ(X, 0) + deg(f ) − 1.
The following result presents a theorem like theorem 1.1 for the Milnor number of
f : (X, 0) → C.
Theorem 4.1. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous curve and let f : (X, 0) →
C be a finite map germ. Let ft be a deformation of f as in (1). Then µ(ft ) is constant if
and only if ft |Xj is a non-negative deformation of f |Xj , for each branch Xj of X.
Proof. By (3), we have
µ(ft ) = µ(X, 0) + deg(ft ) − 1.
Hence, µ(ft ) is constant if and only if deg(ft ) is constant. Assume that (X, 0) has r
branches X1 , . . . , Xr . For each j = 1, . . . , r we choose a point (a1j , . . . , anj ) ∈ Xj not zero
(in such a way that f (a1j , . . . , anj ) is also different from zero by the finiteness of f ). We
define γj : C → Cn , γj (s) = (a1j sw1 , . . . , anj swn ), where (w1 , . . . , wn ) are the weights of
(X, 0). Since γj (C) ⊂ Xj and γj is injective (gcd(w1 , . . . , wn ) = 1), we deduce that γj in
fact parametrizes Xj .
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J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
We see that each branch Xj is also weighted homogenoeus and denote dj = wt(in(f |Xj )),
the degree of the initial part of each f |Xj . Then, we have
deg(ft |Xj ) = m0 (ft (γj (s)))
wn
w1
= m0 (f (a1j s , . . . , anj s ) +
k
X
σi (t)αi (a1j sw1 , . . . , anj swn ))
i=1
= m0 (fdj (a1j s , . . . , anj s ) + fdj +1 (a1j sw1 , . . . , anj swn ) + · · · +
wn
w1
flj (a1j sw1 , . . . , anj swn ) +
k
X
αi (t)αi (a1j sw1 , . . . , anj swn ))
i=1
dj
= m0 (s fdj (a1j , . . . , anj ) + sdj +1 fdj +1 (a1j , . . . , anj ) + · · · +
lj
s flj (a1j , . . . , anj ) +
k
X
σi (t)αi (a1j sw1 , . . . , anj swn ))
i=1
where m0 (p) denotes the multiplicity (i.e., the order) of a function germ p.
We have deg(f |Xj ) = dj and moreover, deg(f |Xj ) = deg(ft |Xj ) if and only if
m0 (αi (a1j sw1 , . . . , anj swn )) ≥ dj ,
for all i such that αi 6= 0, that is, if and only if wt(in(αi )) ≥ dj , for all i such that αi 6= 0.
Furthermore,
r
X
deg(ft ) =
deg(ft |Xj )
j=1
and deg(ft |Xj ) is upper semicontinuous. Therefore, deg(ft ) is constant if and only if
deg(ft |Xj ) is constant for every j. That is, µ(ft ) is constant if and only if ft |Xj is a
non-negative deformation of f |Xj , for each j = 1, . . . , r.
It follows from the results of [20] that µ(ft ) is constant if and only if the family ft is
good and topologically trivial (see [20] for the precise definition of both concepts).
We remark that if a deformation of a map germ f : (X, 0) → C is non-negative it is
not necessarily non-negative in each branch, as we can see in the following example.
Example 4.2. Let (X, 0) be the curve in C2 defined as the zeros of the map φ(x, y) = xy
and let f : (X, 0) → C be the map germ f (x, y) = x3 + y 5 . We define the deformation
ft (x) = x3 + y 5 + ty 4 and we note that, considering X weighted homogeneous with
weights (1, 1), ft is a non-negative deformation of f , but ft |x=0 is not. In this case, we
have, µ(f ) = 8 but µ(ft ) = 7 for t 6= 0.
If, in the previous theorem, f : (X, 0) → C is a semi-weighted homogeneous map germ,
and ft (x) = fd (x) + t[fd+1 (x) + · · · + fl (x)], we have that ft is a non-negative deformation
of fd . Hence, we conclude the following result.
Corollary 4.3. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous curve and let f : (X, 0) →
C be a semi-weighted homogeneous germ. Then µ(f ) = µ(in(f )).
Another case for which the Milnor number of a germ f : (X, 0) → C with an isolated
singularity is defined is when (X, 0) is an ICIS. Assume X = φ−1 (0), where φ : (Cn , 0) →
(Cp , 0) is a holomorphic map germ. Then, we call Milnor number of f to the number:
On
µ(f ) = dimC
,
hφ, J(φ, f¯)i
NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES
7
where f¯ : (Cn , 0) → C is any holomorphic extension of f and J(φ, f¯) is the ideal generated
by the maximal minors of the Jacobian matrix of (φ, f¯). The Lê-Greuel Formula [13] can
be expressed as:
µ(f ) = µ(X, 0) + µ(f −1 (0), 0).
In the following theorem, we present a result analogous to theorem 1.1 for this case.
Theorem 4.4. Let (X, 0) ⊂ (Cn , 0) be a weighted homogeneous ICIS and let f : (X, 0) →
C be a weighted homogeneous germ with an isolated singularity and let ft be a deformation
of f as in (1). If ft is non-negative, then µ(ft ) is constant.
Proof. By the Lê-Greuel formula, we have
µ(ft ) = µ(X, 0) + µ(ft−1 (0), 0).
We only need to show that µ(ft−1 (0), 0) is constant. Moreover, if ft is a non-negative
deformation of f , then we can choose polynomial extensions f¯t : (Cn , 0) → C of ft and
f¯ : (Cn , 0) → C of f such that f¯t is a non-negative deformation of f¯.
On the other hand, if (X, 0) is defined by X = φ−1 (0) with φ : (Cn , 0) → (Cp , 0) then
−1
ft (0) is an ICIS defined by ft−1 (0) = (φ, f¯t )−1 (0). Therefore, it is enough to show that
for a non-negative deformation of a map germ which defines an ICIS, the family of ICIS
has constant Milnor number.
Let ψ = (ψ1 , . . . , ψp ) : (Cn , 0) → Cp be a weighted homogeneous map germ defining
an ICIS (V, 0) = (ψ −1 (0), 0) and let ψt = (ψ1t , . . . , ψpt ) be a non-negative deformation of
Plj j
σi (t)αij (x). We denote Vt = ψt−1 (0). We will proceed by
ψ where ψjt (x) = ψj (x) + i=1
finite induction over p. If p = 1, the result is equivalent to theorem 1.1. Assume that the
result is true for p − 1. By the Lê-Greuel formula,
On
µ(Vt ) = dimC
− µ(Yt , 0),
It
where It is the ideal in On defined by It = hψ1t , . . . , ψp−1t , J(ψ1t , . . . , ψpt )i and Yt is given
by the zeros of (ψ1t , . . . , ψp−1,t ). Since ψt is a non-negative deformation of ψ, we have that
wt(αij ) ≥ wt(ψj ), for all j = 1, . . . , p.
By the induction hypothesis, µ(Yt , 0) is constant. Moreover, it is not difficult to show
that each function in {ψ1t , . . . , ψp−1t , J(ψ1t , . . . , ψpt )} is a non-negative deformation of its
corresponding initial term. Since I0 is weighted homogeneous and dimC On /I0 < ∞ we
have v(I0 ) = {0}, where v(I0 ) denotes the set of zeros of the ideal I0 . Then, it follows
from the results of [6] that v(It ) = {0} also for any t.
On the other hand, since On+1 /It is Cohen-Macaulay, we can apply the principle of
conservation of number
X
On
On,x
On
dimC
=
dimC
= dimC
.
I0
It,x
It
x∈v(It )
Therefore, µ(Vt , 0) = µ(V, 0), for any t.
Remark 4.5. From the proof of the previous theorem, we have that if (X, 0) is the ICIS
defined by the zero set of the weighted homogeneous map germ φ : (Cn , 0) → Cp , φt is
a non-negative deformation of φ and (Xt , 0) is the ICIS defined by φt , then the Milnor
number of (Xt , 0) is constant. However, it is not clear for us what would be a non-negative
deformations of a weighted homogeneous ICIS because it would depend on the generators
of the ideal defining the ICIS.
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J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
5. Some Remarks
In this section the germs are weighted homogeneous with weights (w1 , . . . , wn ). In [16],
it is shown that if f : (Cn , 0) → C is weighted homogeneous germ of degree d with an
isolated singularity, then its Milnor number is
d − w1
d − wn
...
.
µ(f ) =
w1
wn
In [18], we show a similar result for the Bruce-Roberts number. If X = φ−1 (0) is
a weighted homogeneous hypersurface with isolated singularity with wt(φ) = d2 and
f : (Cn , 0) → C is weighted homogeneous RX -finitely determined germ with wt(f ) = d1 ,
then
n
X
d1
µBR (f, X) =
(d1 − w1 ) . . . (d1 − wj−1 )(d2 − wj+1 ) . . . (d2 − wn ).
w1 , . . . , wn j=1
In the following remark we exhibit an analogous formula for the Milnor number of a
function on a curve.
Remark 5.1. Let (X, 0) be a curve defined by the zeros of a weighted homogeneous map
φ = (φ1 , . . . , φn−1 ) : (Cn , 0) → (Cn−1 , 0) with wt(φi ) = di . Let f : (X, 0) → C be a finite
weighted homogeneous function with wt(f ) = dn . Then,
µ(f ) =
d1 . . . dn−1 (d1 + d2 + · · · + dn − w1 − w2 − · · · − wn )
.
w1 . . . wn
In fact, since (X, 0) is an ICIS, we have
µ(f ) = µ(X, 0) + µ(f −1 (0), 0) = dimC On
.
φ1 , . . . , φn−1 , J(f¯, φ)
where J(f¯, φ) is weighted homogenous of degree d1 + d2 + · · · + dn − w1 − w2 − · · · − wn .
Thus, the result follows from Bezout theorem.
By using the above remark, we can write the Milnor number of an 1-dimensional ICIS
in terms of the weights and the degrees. We remark that this result was already obtained
in different ways in [2] and in [10].
Remark 5.2. Let (X, 0) be a curve defined by the zeros of a weighted homogeneous map
φ = (φ1 , . . . , φn−1 ) : (Cn , 0) → (Cn−1 , 0) wt(φi ) = di . Then,
µ(X, 0) =
d1 . . . dn−1 (d1 + d2 + · · · + dn−1 − w1 − w2 − · · · − wn )
+ 1.
w1 . . . wn
In fact, take any weighted homogeneous finite function f : (X, 0) → (C, 0). By (3), we
have
µ(X, 0) = µ(f ) − deg(f ) + 1.
To compute the degree of f we use again Bezout theorem:
deg(f ) = dimC
OX,0
On
d1 . . . dn−1 dn
= dimC
=
.
hf i
hφ1 , . . . , φn−1 , f i
w1 . . . wn
NON-NEGATIVE DEFORMATIONS OF WEIGHTED HOMOGENEOUS SINGULARITIES
9
Hence,
d1 . . . dn−1 (d1 + d2 + · · · + dn−1 + dn − w1 − w2 − · · · − wn ) − d1 . . . dn−1 dn
+1
w1 . . . wn
d1 . . . dn−1 (d1 + d2 + · · · + dn−1 − w1 − w2 − · · · − wn )
=
+ 1.
w1 . . . wn
µ(X, 0) =
Example 5.3. Let (X, 0) = (φ−1 (0), 0), with φ : (C3 , 0) → C2 defined by φ(x, y, z) =
(x2 − y 3 − z 2 , xy 3 − z 3 ), and let f : (X, 0) → C be the function f (x, y, z) = xz − y 3 . We
have that f and φ are weighted homogeneous with weights (3, 2, 3) and degrees (6, 9) and
(6), respectively.
Then,
6 · 9 · (6 + 9 + 6 − 3 − 2 − 3)
µ(f ) =
= 39,
3·2·3
and
6 · 6 · (6 + 9 − 3 − 2 − 3)
+ 1 = 22.
µ(X, 0) =
3·2·3
Remark 5.4. If (X, 0) is a weighted homogeneous curve with r branches and f : (X, 0) →
C is weighted homogeneous on each branch of X with weighted degree dj , then from the
proof of the theorem 4.1, deg(f ) = d1 + · · · + dr , hence
µ(f ) = µ(X) + d1 + · · · + dr − 1.
Inspired by the results of this section it is natural to ask if the Milnor number of a
weighted homogeneous curve depends only on its weights and degrees. In a forthcoming
paper, we will consider this question for weighted homogeneous IDS.
Remark 5.5. Let f : (Cn , 0) → C be a polynomial function with isolated singularity. We
write f = fd + · · · + fl , where fi is weighted homogeneous of degree i. In [7], it is proved
that
(d − w1 ) . . . (d − wn )
µ(f ) ≥
w1 . . . wn
and the equality holds if and only if f is semi-weighted homogeneous.
Here we show an analogous result for a finite polynomial function f : (X, 0) → C on a
weighted homogenous curve (X, 0) ⊂ (Cn , 0). We write f = fd + · · · + fl , where each fi
is weighted homogeneous of degree i. We have
µ(f ) ≥ µ(X, 0) + dr − 1
where r is the number of branches of X. The equality holds if and only if f restricted to
each branch of X is semi-weighted homogeneous.
In fact, let Xj , j = 1, . . . , r be the branches of X. We parametrize each Xj by γj (s) =
(a1j sw1 , . . . , anj swj ) as in the proof of the theorem 4.1. Then, for each j,
deg(f |Xj ) = m0 (f (γj (s)))
= m0 (f (a1j sw1 , . . . , anj swn ))
= m0 (fd (a1j sw1 , . . . , anj swn ) + · · · + fl (a1j sw1 , . . . , anj swn ))
= m0 (sd fd (a1j , . . . , anj ) + · · · + sk fl (a1j , . . . , anj )
≥ d,
and it is equal to d if and only if fd restricted to each branch is finite.
10
J. J. NUÑO-BALLESTEROS, B. ORÉFICE-OKAMOTO, J. N. TOMAZELLA
Hence,
µ(f ) = µ(X, 0) + deg(f ) − 1
r
X
= µ(X, 0) +
deg(f |Xj ) − 1
j=1
≥ µ(X, 0) + dr − 1
and the equality holds if and only if f restricted to each branch is semi-weighted homogeneous.
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Departament de Geometria i Topologia, Universitat de València, Campus de Burjassot,
46100 Burjassot SPAIN
E-mail address: [email protected]
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676,
13560-905, São Carlos, SP, BRAZIL
E-mail address: [email protected]
Departamento de Matemática, Universidade Federal de São Carlos, Caixa Postal 676,
13560-905, São Carlos, SP, BRAZIL
E-mail address: [email protected]